This invention relates generally to the field of reflective surfaces that are capable of providing a wide field of view, and, in particular, to catadioptric sensors that are capable of providing a wide field of view.
The ability of a curved mirror to increase one""s field of view is familiar from their use in stores for security purposes and as rear-view mirrors on automobiles. In both cases, the goal is to allow an observer to see more of a scene than would otherwise be visible.
The xe2x80x9cdistortionxe2x80x9d typically seen in curved mirrors depends upon the shape of the mirror. Here, the word distortion implies that there is something unnatural or wrong with the projections obtained from these mirrors. A more accurate way to say that a mirror distorts the scene is simply to say that the projection is not a perspective projection. A perspective projection is formed by tracing a line from the image plane through a point (called the focal point or center of projection) until it touches an object in the scene. This is how a pinhole camera forms images.
Historically, it was only possible to construct mirrors in spherical or parabolic shapes. These shapes were appropriate for the traditional applications, such as astronomy. In recent years though, it has become possible through computer numerically controlled machining to create parts of almost any given mathematical shape. Consequently, it is now possible to make mirrors with an exactly prescribed geometry.
These developments are applicable to some parts of computer vision and related applications (e.g. robot control). Computer vision research has been dominated for decades by the traditional lens-CCD (charge-coupled device) sensor paradigm, and this paradigm can now be extended due to recent technological advances.
Recently, many researchers in the robotics and vision community have considered visual sensors that are able to obtain wide fields of view. Such devices are the natural solution to various difficulties encountered with conventional imaging systems.
The two most common means of obtaining wide fields of view are fish-eye lenses and reflective surfaces, also known as catoptrics. When catoptrics are combined with conventional lens systems, known as dioptrics, the resulting sensors are known as catadioptrics. The possible uses of these systems include applications such as robot control and surveillance. The present application is directed to catadioptric based sensors.
In the past few years, there has been a tremendous increase in research on the design and applications of catadioptric based sensors. Much of this work has been focused on designing sensors with a panoramic or wide field of view.
In S. Nayar, xe2x80x9cCatadioptric Omnidirectional Cameraxe2x80x9d, Proc. Computer Vision Pattern Recognition, pages 482-88 (1997), Nayar describes a true omni-directional sensor. In this case, the goal was to reconstruct perspective views. This sensor uses a parabolic mirror, which is essentially the only shape from which one can do a perspective unwarping of the image when using a camera that is well modeled by an orthographic projection (see S. Baker and S. Nayar, xe2x80x9cA Theory of Catadioptric Image Formationxe2x80x9d, Proc. International Conference on Computer Vision, pages 35-42 (1998).
A different use of catadioptric sensors is an application of C. Pegard and E. Mouaddib, xe2x80x9cA Mobile Robot Using a Panoramic viewxe2x80x9d, Proc. IEEE Conference on Robotics and Automation, pages 89-94 (1996). In this case, a conical mirror is used to estimate a robot""s pose. This is done using vertical lines in the world as landmarks, which appear as radial lines in the image. If the positions of these landmarks are known, then they can be used to estimate the robot""s pose. In contrast to xe2x80x9cCatadioptric Omnidirectional Cameraxe2x80x9d cited above, in this work, the authors use their device as a 2D sensor.
Navigation and map building with a mobile robot using a conical mirror is considered in Y. Yagi, S. Nishizawa, and S. Tsuji, xe2x80x9cMap-Based Navigation for a Mobile Robot with Omnidirectional Image Sensorxe2x80x9d, Trans. on Robotics and Automation I, pgs. 1:634-1:648 (1995) and Y Yagi, S. Kawato, and S. Tsuji, xe2x80x9cReal-Time Omnidirectional Image Sensor (Copis) for Vision-Guided Navigationxe2x80x9d Trans. on Robotics and Automation, 10:11-10:22 (1994).
In J. Chahl and M. Srinivasan, xe2x80x9cRange Estimation with a Panoramic Sensorxe2x80x9d, J. Optical Soc. Amer. A, pgs. 14:2144-14:2152 (1997), the authors describe a means of estimating range by moving a panoramic sensor, based on the fact that the local distortion of the image is range dependent. This method, which gives a range estimate in every azimuthal direction, is implemented using a conical mirror.
The work most related to the present invention is described in T. Conroy and J. Moore, xe2x80x9cResolution Invariant Surfaces for Panoramic Vision Systemsxe2x80x9d, Proc. International Conference on Computer Vision, pgs. 392-97 (1999) and in J. Chahl and M. Srinivasan, xe2x80x9cReflective surfaces for panoramic imagingxe2x80x9d, Applied Optics, 36:8275-8285, 1997. In T. Conroy and J. Moore, xe2x80x9cResolution invariant surfaces for panoramic vision systemsxe2x80x9d, Proc. International Conference on Computer Vision, pgs. 392-97 (1999), the authors derive a family of mirrors for which the resolution in the image is invariant to changes in elevation. In J. Chahl and M. Srinivasan. xe2x80x9cReflective Surfaces for Panoramic Imagingxe2x80x9d, Applied Optics, pgs. 36:8275-36:8285 (1997), the authors exhibit a family of reflective surfaces that preserve a linear relationship between the angle of incidence of light onto a surface and the angle of reflection onto the imaging device.
Accordingly, it is a general object of the present invention to provide a sensor that is capable of providing a wide field of view with minimal distortion.
It is a further object of the present invention to provide a catadioptric sensor that is capable of providing a wide field of view with minimal distortion.
It is a further object of the present invention to provide a catadioptric sensor that is capable of providing a wide field of view with minimal distortion for use in the robotics and vision community.
It is still a further object of the present invention to provide a catadioptric sensor that is capable of providing a wide field of view with minimal distortion that does not require a mechanical pan-tilt system.
It is yet another object of the present invention to provide a catadioptric sensor that is capable of providing a wide field of view with minimal distortion that does not require computerized unwarping.
Other objects and many attendant features of this invention will become readily appreciated as the same becomes better understood by reference to the following detailed description when considered in connection with the accompanying drawings.
The present invention is directed to two families of reflective surfaces that are capable of providing a wide field of view, and yet still approximate a perspective projection to a high degree. These surfaces are derived by considering a plane perpendicular to the axis of a surface of revolution and finding the equations governing the distortion of the image of the plane in this surface. This relation is then viewed as a differential equation and the distortion term is prescribed to be linear. By choosing appropriate initial conditions for the differential equation and solving it numerically, the surface shape is derived and a precise estimate as to what degree the resulting sensor can approximate a perspective projection is obtained. Thus, these surfaces act as computational sensors, allowing for a wide-angle perspective view of a scene without processing the image in software. The applications of such a sensor are numerous, including surveillance, robotics and traditional photography.
These and other objects of this invention are achieved by providing a mirror for use in a catadioptric system which is a substantially circular mirror having a rotationally symmetric cross section by a set of points substantially close to a curve satisfied by a differential equation as follows:             2      ⁢                        F          xe2x80x2                ⁡                  (          x          )                            1      -                                    F            xe2x80x2                    ⁡                      (            x            )                          2              =                    d        ⁡                  (          x          )                    -      x              F      ⁡              (        x        )            
where x is the radius of the mirror and F(x) is the cross-sectional shape, and d is linear. Alternatively, a mirror for use in a catadioptric system is provided which is a substantially circular mirror having a rotationally symmetric cross section determined by a differential equation as follows:                     x        f            +                        2          ⁢                                    F              xe2x80x2                        ⁡                          (              t              )                                                1          -                                                    F                xe2x80x2                            ⁡                              (                t                )                                      2                                      1      -                        x          f                ⁢                              2            ⁢                                          F                xe2x80x2                            ⁡                              (                t                )                                                          1            -                                                            F                  xe2x80x2                                ⁡                                  (                  t                  )                                            2                                            =                    d        ⁡                  (          x          )                    -      t              F      ⁡              (        t        )            
where x=ft/(F(t)xe2x88x92fxe2x88x92h), f is the focal length, h is the height above an object plane, and F(t) is the cross-sectional shape.